A minimization problem

Given [itex]R_1, R_2, R_3\in\mathbb{R}^{N\times N}[/itex] are symmetric matrices with rank [itex]p<N[/itex]. Their SVD are [itex]U_1\Sigma_1 U_1^T[/itex], [itex]U_2\Sigma_2 U_2^T[/itex] and [itex]U_3\Sigma_3 U_3^T[/itex], respectively. I want to find a rank [itex]p[/itex] matrix [itex]V[/itex] such that